Most computers built after 1980 should be able to do it with some tinkering. Pretty much anything built after 1990 should be fairly trivial to get online.
If you want something somewhat usable, then Lynx/Links running on whatever a modern Linux distro can boot on should suffice.
I used a modem to get online with my Amiga 600 (7 MHz 68000, 2 + 4 MB RAM) in the 90's. It was perfectly possible to use things like IRC, eMule and a WWW browser on it, but the modern web will be quite unusable with aMosaic...